General One-Dimensional Advective-Diffusive Transport

Description

Estimates the transport term (i.e. the rate of change of a concentration due to diffusion and advection) in a one-dimensional model of a liquid (volume fraction constant and equal to one) or in a porous medium (volume fraction variable and lower than one).

The interfaces between grid cells can have a variable cross-sectional area, e.g. when modelling spherical or cylindrical geometries (see example).

Usage

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tran.1D(C, C.up = C[1], C.down = C[length(C)],
        flux.up = NULL, flux.down = NULL, 
        a.bl.up = NULL, a.bl.down = NULL, 
        D = 0, v = 0, AFDW = 1, VF = 1, A = 1, dx,
        full.check = FALSE, full.output = FALSE)

Arguments

C

concentration, expressed per unit of phase volume, defined at the centre of each grid cell. A vector of length N [M/L3]

C.up

concentration at upstream boundary. One value [M/L3]

C.down

concentration at downstream boundary. One value [M/L3]

flux.up

flux across the upstream boundary, positive = INTO model domain. One value, expressed per unit of total surface [M/L2/T]. If NULL, the boundary is prescribed as a concentration or a convective transfer boundary.

flux.down

flux across the downstream boundary, positive = OUT of model domain. One value, expressed per unit of total surface [M/L2/T]. If NULL, the boundary is prescribed as a concentration or a convective transfer boundary.

a.bl.up

convective transfer coefficient across the upstream boundary layer. Flux = a.bl.up*(C.up-C0). One value [L/T]

a.bl.down

convective transfer coefficient across the downstream boundary layer (L). Flux = a.bl.down*(CL-C.down). One value [L/T]

D

diffusion coefficient, defined on grid cell interfaces. One value, a vector of length N+1 [L2/T], or a 1D property list; the list contains at least the element int (see setup.prop.1D) [L2/T]

v

advective velocity, defined on the grid cell interfaces. Can be positive (downstream flow) or negative (upstream flow). One value, a vector of length N+1 [L/T], or a 1D property list; the list contains at least the element int (see setup.prop.1D) [L/T]

AFDW

weight used in the finite difference scheme for advection, defined on grid cell interfaces; backward = 1, centred = 0.5, forward = 0; default is backward. One value, a vector of length N+1, or a 1D property list; the list contains at least the element int (see setup.prop.1D) [-]

VF

Volume fraction defined at the grid cell interfaces. One value, a vector of length N+1, or a 1D property list; the list contains at least the elements int and mid (see setup.prop.1D) [-]

A

Interface area defined at the grid cell interfaces. One value, a vector of length N+1, or a 1D grid property list; the list contains at least the elements int and mid (see setup.prop.1D) [L2]

dx

distance between adjacent cell interfaces (thickness of grid cells). One value, a vector of length N, or a 1D grid list containing at least the elements dx and dx.aux (see setup.grid.1D) [L]

full.check

logical flag enabling a full check of the consistency of the arguments (default = FALSE; TRUE slows down execution by 50 percent)

full.output

logical flag enabling a full return of the output (default = FALSE; TRUE slows down execution by 20 percent)

Details

The boundary conditions are either

  • (1) zero-gradient.

  • (2) fixed concentration.

  • (3) convective boundary layer.

  • (4) fixed flux.

The above order also shows the priority. The default condition is the zero gradient. The fixed concentration condition overrules the zero gradient. The convective boundary layer condition overrules the fixed concentration and zero gradient. The fixed flux overrules all other specifications.

Ensure that the boundary conditions are well defined: for instance, it does not make sense to specify an influx in a boundary cell with the advection velocity pointing outward.

Transport properties:

The diffusion coefficient (D), the advective velocity (v), the volume fraction (VF), the interface surface (A), and the advective finite difference weight (AFDW) can either be specified as one value, a vector or a 1D property list as generated by setup.prop.1D.

When a vector, this vector must be of length N+1, defined at all grid cell interfaces, including the upper and lower boundary.

The finite difference grid (dx) is specified either as one value, a vector or a 1D grid list, as generated by setup.grid.1D.

Value

dC

the rate of change of the concentration C due to transport, defined in the centre of each grid cell. The rate of change is expressed per unit of phase volume [M/L3/T]

C.up

concentration at the upstream interface. One value [M/L3] only when (full.output = TRUE)

C.down

concentration at the downstream interface. One value [M/L3] only when (full.output = TRUE)

dif.flux

diffusive flux across at the interface of each grid cell. A vector of length N+1 [M/L2/T] only when (full.output = TRUE)

adv.flux

advective flux across at the interface of each grid cell. A vector of length N+1 [M/L2/T] only when (full.output = TRUE)

flux

total flux across at the interface of each grid cell. A vector of length N+1 [M/L2/T]. only when (full.output = TRUE)

flux.up

flux across the upstream boundary, positive = INTO model domain. One value [M/L2/T]

flux.down

flux across the downstream boundary, positive = OUT of model domain. One value [M/L2/T]

Note

The advective equation is not checked for mass conservation. Sometimes, this is not an issue, for instance when v represents a sinking velocity of particles or a swimming velocity of organisms. In others cases however, mass conservation needs to be accounted for. To ensure mass conservation, the advective velocity must obey certain continuity constraints: in essence the product of the volume fraction (VF), interface surface area (A) and advective velocity (v) should be constant. In sediments, one can use setup.compaction.1D to ensure that the advective velocities for the pore water and solid phase meet these constraints.

In terms of the units of concentrations and fluxes we follow the convention in the geosciences. The concentration C, C.up, C.down as well at the rate of change of the concentration dC are always expressed per unit of phase volume (i.e. per unit volume of solid or liquid).

Total concentrations (e.g. per unit volume of bulk sediment) can be obtained by multiplication with the appropriate volume fraction. In contrast, fluxes are always expressed per unit of total interface area (so here the volume fraction is accounted for).

Author(s)

Filip Meysman <filip.meysman@nioz.nl>, Karline Soetaert <karline.soetaert@nioz.nl>

References

Soetaert and Herman (2009). A practical guide to ecological modelling - using R as a simulation platform. Springer

See Also

tran.volume.1D for a discretisation the transport equation using finite volumes.

tran.2D, tran.3D

advection.1D, for more sophisticated advection schemes

Examples

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## =============================================================================
## EXAMPLE 1: O2 and OC consumption in sediments
## =============================================================================

# this example uses only the volume fractions 
# in the reactive transport term

#====================#
# Model formulation  #
#====================#

# Monod consumption of oxygen (O2)

O2.model <- function (t = 0, O2, pars = NULL) {

  tran <- tran.1D(C = O2, C.up = C.ow.O2, D = D.grid, 
                  v = v.grid, VF = por.grid, dx = grid)$dC
  reac <- - R.O2*(O2/(Ks+O2))
  return(list(dCdt = tran + reac))
}

# First order consumption of organic carbon (OC)

OC.model <- function (t = 0, OC, pars = NULL) {

  tran <- tran.1D(C = OC, flux.up = F.OC, D = Db.grid,
                  v = v.grid, VF = svf.grid, dx = grid)$dC
  reac <- - k*OC
  return(list(dCdt = tran + reac))
}

#======================#
# Parameter definition #
#======================#

# Parameter values

F.OC    <- 25    # input flux organic carbon [micromol cm-2 yr-1]
C.ow.O2 <- 0.25  # concentration O2 in overlying water [micromol cm-3]
por     <- 0.8   # porosity
D       <- 400   # diffusion coefficient O2 [cm2 yr-1]
Db      <- 10    # mixing coefficient sediment [cm2 yr-1]
v       <- 1     # advective velocity [cm yr-1]
k       <- 1     # decay constant organic carbon [yr-1]
R.O2    <- 10    # O2 consumption rate [micromol cm-3 yr-1]
Ks      <- 0.005 # O2 consumption saturation constant 

# Grid definition

L <- 10   # depth of sediment domain [cm]
N <- 100  # number of grid layers
grid <- setup.grid.1D(x.up = 0, L = L, N = N)

# Volume fractions 

por.grid <- setup.prop.1D(value = por, grid = grid)
svf.grid <- setup.prop.1D(value = (1-por), grid = grid)
D.grid   <- setup.prop.1D(value = D, grid = grid)
Db.grid  <- setup.prop.1D(value = Db, grid = grid)
v.grid   <- setup.prop.1D(value = v, grid = grid)

#====================#
# Model solution     #
#====================#

# Initial conditions + simulation O2

yini <- rep(0, length.out = N) 
O2   <- steady.1D(y = yini, func = O2.model, nspec = 1)

# Initial conditions + simulation OC

yini <- rep(0, length.out = N) 
OC   <- steady.1D(y = yini, func = OC.model, nspec = 1)

# Plotting output, using S3 plot method of package rootSolve"

plot(O2, grid = grid$x.mid, xyswap = TRUE, main = "O2 concentration", 
     ylab = "depth [cm]", xlab = "", mfrow = c(1,2), type = "p", pch = 16)

plot(OC, grid = grid$x.mid, xyswap = TRUE, main = "C concentration", 
     ylab = "depth [cm]", xlab = "", mfrow = NULL)

## =============================================================================
## EXAMPLE 2: O2 in a cylindrical and spherical organism
## =============================================================================

# This example uses only the surface areas 
# in the reactive transport term

#====================#
# Model formulation  #
#====================#

# the numerical model - rate of change = transport-consumption
Cylinder.Model <- function(time, O2, pars)
  return (list(
    tran.1D(C = O2, C.down = BW, D = Da, A = A.cyl, dx = dx)$dC - Q
    ))

Sphere.Model <- function(time, O2, pars)
  return (list(
    tran.1D(C = O2, C.down = BW, D = Da, A = A.sphere, dx = dx)$dC - Q
    ))

#======================#
# Parameter definition #
#======================#

# parameter values

BW     <- 2      # mmol/m3,  oxygen conc in surrounding water
Da     <- 0.5    # cm2/d     effective diffusion coeff in organism
R      <- 0.0025 # cm        radius of organism
Q      <- 250000 # nM/cm3/d  oxygen consumption rate/ volume / day
L      <- 0.05   # cm        length of organism (if a cylinder)

# the numerical model

N  <- 40                              # layers in the body
dx <- R/N                             # thickness of each layer
x.mid <- seq(dx/2, by = dx, length.out = N) # distance of center to mid-layer
x.int <- seq(0, by = dx, length.out = N+1)  # distance to layer interface

# Cylindrical surfaces
A.cyl   <- 2*pi*x.int*L  # surface at mid-layer depth

# Spherical surfaces
A.sphere <- 4*pi*x.int^2 # surface of sphere, at each mid-layer

#====================#
# Model solution     #
#====================#

# the analytical solution of cylindrical and spherical model
cylinder <- function(Da, Q, BW, R, r)  BW + Q/(4*Da)*(r^2-R^2)
sphere   <- function(Da, Q, BW, R, r)  BW + Q/(6*Da)*(r^2-R^2)

# solve the model numerically for a cylinder
O2.cyl <- steady.1D (y = runif(N), name = "O2", 
      func = Cylinder.Model, nspec = 1, atol = 1e-10)

# solve the model numerically for a sphere
O2.sphere <- steady.1D (y = runif(N), name = "O2", 
      func = Sphere.Model, nspec = 1, atol = 1e-10)

#====================#
# Plotting output    #
#====================#
# Analytical solution - "observations"
Ana.cyl   <- cbind(x.mid, O2 = cylinder(Da, Q, BW, R, x.mid))
Ana.spher <- cbind(x.mid, O2 = sphere(Da, Q, BW, R, x.mid))

plot(O2.cyl, O2.sphere, grid = x.mid, lwd = 2, lty = 1, col = 1:2, 
     xlab = "distance from centre, cm", 
     ylab = "mmol/m3", main = "tran.1D",
     sub = "diffusion-reaction in a cylinder and sphere",
     obs = list(Ana.cyl, Ana.spher), obspar = list(pch = 16, col =1:2))
     
legend ("topleft", lty = c(1, NA), pch = c(NA, 18),
        c("numerical approximation", "analytical solution"))
legend ("bottomright", pch = 16, lty = 1, col = 1:2,
        c("cylinder", "sphere"))

## =============================================================================
## EXAMPLE 3: O2 consumption in a spherical aggregate
## =============================================================================

# this example uses both the surface areas and the volume fractions
# in the reactive transport term

#====================#
# Model formulation  #
#====================#

Aggregate.Model <- function(time, O2, pars) {

  tran <- tran.1D(C = O2, C.down = C.ow.O2,
                  D = D.grid, A = A.grid,
                  VF = por.grid, dx = grid )$dC

  reac <- - R.O2*(O2/(Ks+O2))*(O2>0)
  return(list(dCdt = tran + reac, consumption = -reac))

}

#======================#
# Parameter definition #
#======================#

# Parameters

C.ow.O2 <- 0.25     # concentration O2 water [micromol cm-3]
por     <- 0.8      # porosity
D       <- 400      # diffusion coefficient O2 [cm2 yr-1]
v       <- 0        # advective velocity [cm yr-1]
R.O2    <- 1000000  # O2 consumption rate [micromol cm-3 yr-1]
Ks      <- 0.005    # O2 saturation constant [micromol cm-3]

# Grid definition
R <- 0.025           # radius of the agggregate [cm]
N <- 100             # number of grid layers
grid <- setup.grid.1D(x.up = 0, L = R, N = N)

# Volume fractions 

por.grid <- setup.prop.1D(value = por, grid = grid)
D.grid   <- setup.prop.1D(value = D, grid = grid)

# Surfaces 

A.mid <- 4*pi*grid$x.mid^2  # surface of sphere at middle of grid cells
A.int <- 4*pi*grid$x.int^2  # surface of sphere at interface
A.grid <- list(int = A.int, mid = A.mid)

#====================#
# Model solution     #
#====================#

# Numerical solution: staedy state 

O2.agg <- steady.1D (runif(N), func = Aggregate.Model, nspec = 1,
                     atol = 1e-10, names = "O2")

#====================#
# Plotting output    #
#====================#

par(mfrow = c(1,1))

plot(grid$x.mid, O2.agg$y, xlab = "distance from centre, cm",
     ylab = "mmol/m3",
     main = "Diffusion-reaction of O2 in a spherical aggregate")
legend ("bottomright", pch = c(1, 18), lty = 1, col = "black",
        c("O2 concentration"))

# Similar, using S3 plot method of package rootSolve"
plot(O2.agg, grid = grid$x.mid, which = c("O2", "consumption"),
     xlab = "distance from centre, cm", ylab = c("mmol/m3","mmol/m3/d"))